A Radial Basis Function-Hermite Finite Difference Method for the Two-Dimensional Distributed-Order Time-Fractional Cable Equation

Abstract

In this article, our main objective is to propose a high-order local meshless method for numerical solution of two-dimensional distributed-order time-fractional cable equation on both regular and irregular domains. First, the distribution-order integral is approximated by the Gauss-Legendre quadrature formula, and then a second-order weighted and shifted Grünwald difference (WSGD) scheme is applied to approximate the time Riemann-Liouville derivatives. The stability and convergence analysis of the time-discrete outline are investigated by the energy approach. The spatial dimension of the model is discretized by the fourth-order local radial basis function-Hermite finite difference (RBF-HFD) method. Some numerical experiments are performed on regular and irregular computational domains to verify the ability, efficiency, and accuracy of the proposed numerical procedure. The numerical simulations clearly demonstrate the high accuracy of the provided numerical process in comparison to existing procedures. Finally, it can be concluded that the presented technique is a suitable alternative to the existing numerical techniques for the distributed-order time-fractional cable equation.